Symmetric and Skew Symmetric Matrices
Symmetric Matrix:
A square matrix $A$ is symmetric if it is equal to its transpose: $A={A}^{T}$
 Properties:
 The elements along the main diagonal remain unchanged when transposed.
 ${a}_{ij}={a}_{ji}$ for all $i$ and $j$.
 It is always a square matrix (number of rows = number of columns).
Example of Symmetric Matrix:
$$A=\left[\begin{array}{ccc}{\textstyle 2}& {\textstyle 5}& {\textstyle 7}\\ {\textstyle 5}& {\textstyle 3}& {\textstyle 9}\\ {\textstyle 7}& {\textstyle 9}& {\textstyle 1}\end{array}\right]$$
In this case, $A={A}^{T}$, which makes it a symmetric matrix.
SkewSymmetric Matrix:
A square matrix $A$ is skewsymmetric if it is equal to the negative of its transpose: $A={A}^{T}$
 Properties:
 The elements along the main diagonal are zero.
 ${a}_{ij}={a}_{ji}$ for all $i$ and $j$.
 It is always a square matrix (number of rows = number of columns).
Example of SkewSymmetric Matrix:
$$A=\left[\begin{array}{ccc}{\textstyle 0}& {\textstyle 7}& {\textstyle 3}\\ {\textstyle 7}& {\textstyle 0}& {\textstyle 2}\\ {\textstyle 3}& {\textstyle 2}& {\textstyle 0}\end{array}\right]$$
In this case, $A={A}^{T}$, which makes it a skewsymmetric matrix.
Properties of Symmetric and SkewSymmetric Matrices:

Sum of Symmetric Matrices: The sum of two symmetric matrices is also symmetric. $A+B=C$where $A$, $B$, and $C$ are symmetric matrices.

Sum of SkewSymmetric Matrices: The sum of two skewsymmetric matrices is also skewsymmetric. $A+B=C$ where $A$, $B$, and $C$ are skewsymmetric matrices.

Product with Scalar: Both symmetric and skewsymmetric matrices preserve their property when multiplied by a scalar.